The proposition above is not new, and proofs of some of the statements can be found in [1]} and [2]}.
For completeness, we provide a proof of this proposition using primary decomposition and Hilbert-Samuel multiplicity of an ideal.
In the former, the authors study the cycle decomposition of the intersection of two quadric hypersurfaces to obtain results about existence of rational points on intersection of two quadric hypersurfaces and Châtelet surfaces over number fields.
Our statements here are slightly simpler to state (and slightly different) since in the works above the authors work in the setting of perfect fields, whereas we are concerned with the special case of algebraically closed fields of characteristic zero.
